Reducing GPS orbital and clocks errors using recurrent neural networks

Reducing GPS orbital and clocks errors using recurrent neural networks | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Reducing GPS orbital and clocks errors using recurrent neural networks Akram Abderrahaman Bousnane, Mohamed Cherif Berguig, Hicham Dekkiche This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3881623/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Because satellite orbits and clocks are usually treated as known quantities in GPS positioning models, these errors are one of the most impactful in Global Navigation satellite system (GNSS) positioning, especially in single point positioning mode. In this work we tried to reduce orbit and clock errors using deep learning technique (recurrent neural network) by creating a model that predict the offsets of broadcast ephemeris and clock biases, from precise products. We tried two different methods, the first one is predicting the future offsets for the next + 72 hours basing on both precise and broadcast ephemeris, and the second method is predicting the current offsets using only broadcast ephemeris as inputs. To create the models, we used Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) architectures. These architectures are generally used to solve sequential problems in deep learning. We got results that vary between methods in terms of satellites and time steps. For the first method we get (10-50cm) and (0.5-1.7ns) reduced from (~ 130cm, ~ 14ns), It increases over time steps, and we get a minimum of (44cm) for the second method. Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Introduction GNSS satellites fly in very precise, well-known trajectory (orbit), defined by Keplerian ellipse. However, the gravitational and non-gravitational forces deviate the satellite from its original orbit. Therefore, ground control system continually monitors this orbits changes and provide us with what we call broadcast ephemeris transmitted in navigation message in addition to clocks bias and drift. Even with the correction done by the ground control system. Small errors still exist that can result around ~ 1.0 meter in position (IGS official website) and 5 ns in clock (IGS official website). For some applications, these last-mentioned errors could be considered as significant. The common ways to reduce them is to use precise products provided by GNSS ground control services, like the International GNSS Service (IGS) products. IGS provides three types of products: Ultra rapid, rapid, and final products. The three products are sampled on 15mn time interval with a precision of 0.05m – 0.025m for orbits and ~ 3ns − 75ps for clocks. Another way to compensate those errors in single point positioning mode, is by using SBAS system or Precise Point Positioning (John W et al 2016). This study proposes another way to reduce broadcast ephemeris and clock bias errors by using deep learning techniques. Deep learning (DL) is a very powerful technique in handling time-series data, and it can learn time dependent patterns across multiple models ( Grossi et al 2008 ) . Therefore, DL can be found useful in the discovery of unknown and hidden information in GNSS data. The large quantity of data provided by IGS, made the development of a Dl model possible. The proposed methodology treats the problem in two different methods: one by creating a model that forecasts the future ephemeris and clock offsets based on the past ones. The other is to create a deep learning model that predicts the current orbit errors based on past orbital elements (Keplerian elements). Up to day several research have implemented machine learning algorithms in GNSS field, (Srivani. I et al 2019) (Sivakrishna. K et al 2022) (Orus Perez. R 2019) and others treated ionosphere errors in different ways using deep learning. An ANN (Artificial Neural Network) model was implemented to predict tropospheric wet delay based on meteorological and GNSS data (Selbesoglu. M.O. 2020). Different machine learning techniques were used to detect multipath (Munin. E et al 2020) (Quan. Y 2017). Also, Google in collaboration with the Institute of Navigation (ION) lunched two competitions for 10.000 $ prize pool on Kaggle to compute smartphone’s location down to the decimeter or even centimeter resolution using ML (Google Smartphone Decimeter Challenge 2021 ) (Google Smartphone Decimeter Challenge 2022 ). Recently (He, S 2023) used LSTM to model and predict BDS-3 satellite clock bias. Deep learning is a subfield of machine learning that involves training artificial neural networks with multiple layers. These neural networks are inspired by the structure and function of the human brain and are able to learn and improve their performance over time through a process known as training (Grossi et al 2008). Because we have non-linear and multi-Dimension datasets, the best suitable ML technique in this case is deep learning (Jason Brownlee 2018 ). There are several deep learning architectures that are particularly well suited for time series forecasting such as: Recurrent Neural Networks (RNNs) with its popular variation Long Short-Term Memory (Sepp Hochreiter et al 1997) and (Gated Recurrent Unite) (Cho et al 2014 ), and Convolutional Neural Networks (Convolutional neural networks) (Yamashita et al 2018 ). In this study the three architectures mentioned before are used. However, only LSTM and GRU results will be shared in this paper. Methods Recurrent neural networks Artificial Neural Networks algorithms are a simulation of human brain behavior. This means ANN algorithms learn patterns from massive historical or past data, and apply gained knowledge to predict new results (Grossi et al 2008). Recurrent Neural Networks (RNNs) are a type of artificial neural network that can handle sequential data by allowing information to be passed from one time step to the next, when simple ANN view each observation as independent because the network is not able to retain time steps informations ( Robin M. Schmidt 2019 ). Let's consider a simple RNN with one input, one hidden layer, and one output (Fig. 1). At each time step t, the RNN takes an input vector x(t) and produces an output vector y(t) and a(t). Figure (1) : Structure of simple RNN cell (15). The hidden layer at time step t is denoted by h(t), which is a function of the previous hidden state a(t-1) and the current input x(t). a(t) = f(Wx(t) + Wa(t-1) + Ba) y(t) = g(Wy(t) + By) where f and g are non-linear activation function (tanh, sigmoid, ….), Wx, Wa, Wy are weights for the recurrent, input connections, and output, respectively, and Ba, By are bias vectors. Combining these blocks will give us an RNN model shown below (Fig. 4). To train an RNN, we need to compute the loss at each time step and then back-propagate the errors through time. The loss function is typically defined as the sum of the losses at each time step: L = ∑ L(t) where L(t) is the loss for one time step t. We can then compute the gradient of the loss with respect to the parameters of the RNN using the Back-Propagation Through Time (BPTT) algorithm. The BPTT algorithm works by unrolling the RNN through time and computing the gradients at each time step. The gradients are then accumulated over time and used to update the weights and biases using gradient descent a similar optimization algorithm. However, the problem with RNN is exploding/vanishing gradients during back-propagation step. Because of these problems, RNN cannot capture the relevant information from long-term dependencies because of multiplicative gradient values that can gradually increase/decrease based on the number of hidden layers. Figure (2): Structure of LSTM cell ( Sepp Hochreiter 1997 ). Long-Short-Term-Memory ( Sepp Hochreiter et al 1997 ) LSTM and GRU (Cho et al 2014 ) are a special type of RNNs mainly developed to tackle the main problem of simple RNNs, the problem of vanishing gradients. A common LSTM unit is composed of a cell ( \({C}_{t}\) ), an input gate \(({i}_{t}\) ), an output gate ( \({o}_{t}\) ) and a forget gate \({(f}_{t}\) ). The cell state allows information to flow along the entire chain with only minor linear actions through three gates. The three gates work as filters and control the flow of information and determine which information is kept or disregarded. Forget gates decide what information to discard from a previous state (long-term memory). The value of 1 means keeps the information and the value of 0 means to discard it. Input gates decide which pieces of new information to store in the current state. Output gates control which pieces of information in the current state to output (short-term memory). Both input and output gates use the same system as forget gate bounding values between 0 and 1 by sigmoid. The forms of the equations for an LSTM with a forget gate are (Sepp Hochreiter et al 1997) \({f}_{t}= \sigma ({W}_{f,x}{x}_{t}+{W}_{f,h}{h}_{t-1}+{b}_{f}\) \({i}_{t}= \sigma ({W}_{i,x}{x}_{t}+{W}_{i,h}{h}_{t-1}+{b}_{i})\) $${o}_{t}= \sigma ({W}_{o,x}{x}_{t}+{W}_{o,h}{h}_{t-1}+{b}_{o})$$ $${\stackrel{\sim}{C}}_{t}= tanh({W}_{c,x}{x}_{t}+{W}_{c,h}{h}_{t-1}+{b}_{c})$$ $${C}_{t}= {f}_{t}*{C}_{t-1}+{i}_{t}*{\stackrel{\sim}{C}}_{t}$$ $${h}_{t}= {o}_{t}*\text{t}\text{a}\text{n}\text{h}\left({C}_{t}\right)$$ Figure (3) : Structure of GRU cell (Cho et al 2014 ) GRU is a variant of the LSTM which instead of the input, forget, and output gates in LSTM; GRU has two gates: an update gate z, and a reset gate r. The reset gate ( \({r}_{t}\) ) is responsible for the short-term memory as it decides how much past information is kept and disregarded. Otherwise, the update gate ( \({z}_{t}\) ) is responsible for the long-term memory and is comparable to the LSTM’s forget gate. GRU have variant forms. The common form of GRU is fully gated unit (Fig. 3.) and have the equations ( Cho et al 2014 ): $${r}_{t}= \sigma ({W}_{r,x}{x}_{t}+{W}_{r,h}{h}_{t-1}+{b}_{r})$$ $${z}_{t}= \sigma ({W}_{z,x}{x}_{t}+{W}_{z,h}{h}_{t-1}+{b}_{z})$$ $${C}_{t}= tanh({W}_{c,x}{x}_{t}+{W}_{c,h}({r}_{t}*{h}_{t-1})+{b}_{c})$$ $${h}_{t}= {z}_{t}*{h}_{t-1}+\left(1-{z}_{t}\right)\text{*}{C}_{t}$$ Due to the simpler architecture compared to LSTM, GRUs are computationally more efficient and faster to train as they need less memory. However, as GRUs do not have a separate hidden and cell state gate, they might not be able to consider observations as far into the past as the LSTM. Figure (4): Proposed model for this study. Data description: For this work, two different methods were applied. In which the output for both methods are quite similar (broadcast ephemeris corrections) while the input is different. For method 1 inputs are differences between broadcast and precise ephemeris, while the inputs in second methods are Keplerian element from broadcast ephemeris only. To create our deep learning models, broadcast ephemeris and precise ephemeris were used for all the operational satellites from the lunch date until December 2021. All the files were downloaded from the CDDIS service database (Crustal Dynamics Data Information System Database). Then the ECEF broadcast coordinate was calculated using the algorithm given by (Seeber 2003 ) An interval sample of one hour was used. For each satellite the orbital elements {M, w, e, cuc, cus, crc, crs, Ɵ, i} were taken. For precise ephemeris, the coordinate for each same epoch (1h) was taken. The precise ephemeris products provide the center-of-mass (CoM) coordinates of the respective satellite, while the broadcast ephemeris represents the satellite antenna phase center coordinates. In order to compare the two products, antennas offset correction (AOC) is needed for precise ephemeris. Thus, we used antenna parameters published by National Geospatial-Intelligence Agency (NGA) (NGA 2020). Table 1 Offsets between precise ephemeris and broadcast ephemeris, mean from 1-1-2022 to 15-08-2022 (Test Data) SNV PRN Dx Dy Dz Snv43 PRN13 0.915 0.790 0.886 Snv45 PRN21 2.711 2.712 3.164 Snv48 PRN7 1.516 1.710 1.701 Snv50 PRN5 0.972 0.776 0.899 Snv51 PRN20 0.856 0.930 1.220 Snv52 PRN31 1.220 0.935 1.226 Snv53 PRN17 1.745 1.941 1.892 Snv55 PRN15 1.600 1.370 1.560 Snv56 PRN16 1.526 1.059 1.671 Snv57 PRN29 0.821 1.053 1.312 Snv58 PRN12 0.804 1.580 1.397 Snv59 PRN19 1.561 1.223 1.454 Snv61 PRN2 1.987 2.907 3.128 Snv62 PRN25 1.110 1.102 1.259 Snv63 PRN1 1.220 1.192 1.290 Snv64 PRN30 1.080 0.992 1.321 Snv65 PRN24 1.196 1.165 1.291 Snv66 PRN27 1.082 1.015 1.072 Snv67 PRN6 0.959 0.806 1.108 Snv68 PRN9 0.672 0.841 1.001 Snv69 PRN3 0.904 1.106 1.067 Snv70 PRN32 0.762 0.926 0.860 Snv71 PRN26 0.891 0.739 0.804 Snv72 PRN8 0.997 0.864 0.920 Snv73 PRN10 0.936 1.177 1.259 Snv74 PRN4 0.585 0.586 0.628 Snv75 PRN18 0.695 0.636 0.661 Snv76 PRN23 1.116 1.072 1.054 Snv77 PRN14 0.628 0.771 0.834 Snv78 PRN11 0.628 0.761 0.740 Table (1) shows the offsets of the data that we used to test our models. The average of those offsets is 1.2m. However, we can see that two satellites have large offsets (SNV45, SNV61) both of them are in IIR block. Results and discussion Method 1: The first method is to predict the future offsets based on the past ones. Therefore, we used the last 24h data (both broadcast and precise ephemeris) to predict the next 72h offsets (Many to one model). Fig (5): Method 1 diagram with one input and one output As in fig (5) we can see that, the input victor is a 3D matrix with the shape ( n , 3, 24); where n is the length of the dataset, 3 refers to the three ECEF offset component (Dx, Dy, Dz), and 24 refers to the last 24 epoch. The output is a 2D victor ( n , 3). 1h 2h 6h 12h 24h 48h 72h Snv43 0.10 0.16 0.24 0.29 0.35 0.44 0.47 Snv45 0.32 0.40 0.51 0.62 0.77 0.94 1.28 Snv48 0.13 0.19 0.28 0.33 0.40 0.48 0.51 Snv50 0.10 0.16 0.23 0.27 0.31 0.38 0.44 Snv51 0.12 0.19 0.28 0.31 0.35 0.49 0.49 Snv52 0.10 0.15 0.21 0.25 0.28 0.33 0.38 Snv53 0.15 0.24 0.34 0.41 0.50 0.59 0.66 Snv55 0.12 0.18 0.26 0.30 0.36 0.43 0.49 Snv56 0.10 0.15 0.22 0.26 0.28 0.35 0.43 Snv57 0.10 0.15 0.23 0.27 0.31 0.41 0.40 Snv58 0.13 0.19 0.31 0.35 0.44 0.57 0.57 Snv59 0.11 0.18 0.27 0.30 0.34 0.41 0.45 Snv61 0.21 0.27 0.37 0.44 0.55 0.68 0.81 Snv62 0.10 0.15 0.22 0.25 0.28 0.34 0.38 Snv63 0.11 0.16 0.23 0.27 0.30 0.36 0.40 Snv64 0.11 0.16 0.22 0.26 0.32 0.42 0.41 Snv65 0.11 0.16 0.22 0.26 0.30 0.39 0.43 Snv66 0.10 0.16 0.23 0.27 0.31 0.39 0.42 Snv67 0.10 0.16 0.22 0.25 0.30 0.39 0.44 Snv68 0.10 0.14 0.22 0.26 0.29 0.36 0.40 Snv69 0.11 0.18 0.27 0.35 0.42 0.54 0.56 Snv70 0.10 0.15 0.22 0.25 0.31 0.37 0.42 Snv71 0.09 0.13 0.20 0.23 0.27 0.34 0.38 Snv72 0.13 0.20 0.30 0.36 0.41 0.50 0.54 Snv73 0.12 0.18 0.27 0.30 0.37 0.47 0.50 Snv74 0.24 0.28 0.33 0.39 0.45 0.52 0.59 Snv75 0.23 0.30 0.37 0.41 0.50 0.55 0.62 Snv76 0.30 0.40 0.47 0.53 0.62 0.78 0.94 Snv77 0.29 0.36 0.42 0.48 0.56 0.67 0.78 Snv78 0.30 0.38 0.44 0.50 0.54 0.62 0.71 Table 2: Method 1 mean (Dx, Dy, Dz) results over different epochs Fig (6): Ephemeris model 1 performance through different epochs compared to original data 1h-2h 2h-4h 4h-6h 6h-12h 12h-24h 24h-48h 48h-72h Decay over epochs 0.062 0.078 0.048 0.059 0.090 0.060 0.050 Decay percent 41.98% 37.20% 16.62% 17.67% 22.93% 12.48% 9.12% Decay per hour 0.062 0.039 0.024 0.010 0.008 0.003 0.002 Decay percent per hour 41.98% 18.60% 8.31% 2.95% 1.91% 0.52% 0.38% Table (3): Models performance decay over time From the previous results, Fig (6) and tables (2 and 3), we can see that the original distances are around (1.00-1.50m) except for the new satellites (74-78) (block III) which have distances below 0.90m. The problem with the new satellites is the lack of data which explains the underperformance of the model for these satellites. Also, satellites (61,45) which have an original error of ~2.67m, the model has the worst performance. Without the satellites (i.e., 45, 61, 74-78), an offset of (> 28 cm) will be obtained at 12h and (~ 50cm) at 72h. Taking into the account that, this model is a single model which process all the satellites using GRU architecture. In addition, we may notice that the rate of degradation of model performance gets lower over time. Method 2 Fig (7): Method 2 diagram with one input and one output The second method is an error reduction for broadcast ephemeris. The inputs in this method are Keplerian elements of the last 12 sets of ephemeris (24h) and the output is the correction for the current satellite coordinates (Dx, Dy, Dz). Therefore, no precise ephemeris are required in this model. They are only needed for the training. Also, this method is less accurate than the first, but it turned out to be more practical. Method 2 results LSTM GRU BILSTM BIGRU LSTM (2009 Data) Snv43 0.497 0.515 0.532 0.510 0.531 Snv45 0.687 0.735 0.784 0.709 0.659 Snv48 0.516 0.516 0.556 0.536 0.720 Snv50 0.544 0.565 0.580 0.536 0.544 Snv51 0.833 0.757 0.873 0.810 0.902 Snv52 0.449 0.473 0.472 0.443 0.491 Snv53 0.832 0.888 0.893 0.887 0.833 Snv55 0.625 0.671 0.563 0.635 0.587 Snv56 0.502 0.561 0.562 0.556 0.741 Snv57 0.501 0.556 0.524 0.514 0.640 Snv58 0.645 0.639 0.637 0.642 0.726 Snv59 0.725 0.753 0.644 0.674 0.709 Snv61 0.786 0.698 0.849 0.753 0.657 Snv62 0.561 0.540 0.524 0.533 0.503 Snv63 0.498 0.524 0.479 0.485 0.467 Snv64 0.460 0.463 0.473 0.454 0.457 Snv65 0.517 0.522 0.475 0.473 0.517 Snv66 0.905 0.712 0.966 0.795 1.036 Snv67 0.562 0.601 0.575 0.558 0.651 Snv68 0.453 0.468 0.476 0.470 0.466 Snv69 0.739 0.729 0.723 0.717 0.733 Snv70 0.504 0.492 0.509 0.495 0.481 Snv71 0.460 0.467 0.466 0.492 0.566 Snv72 0.564 0.597 0.569 0.572 0.605 Snv73 0.533 0.594 0.578 0.561 0.567 Snv74 0.479 0.488 0.475 0.465 0.487 Snv75 0.504 0.529 0.494 0.519 0.520 Snv76 0.744 0.770 0.737 0.767 0.688 Snv77 0.594 0.615 0.602 0.598 0.610 Snv78 0.648 0.691 0.647 0.693 0.640 Table (4): Method 2 results for different RNN architectures Dx Dy Dz Correction Correction Percentage Snv43 0.503 0.498 0.490 0.37 42% Snv45 0.653 0.697 0.628 2.20 77% Snv48 0.498 0.528 0.521 1.13 69% Snv50 0.510 0.544 0.553 0.35 39% Snv51 0.722 0.757 0.794 0.24 24% Snv52 0.459 0.433 0.436 0.68 61% Snv53 0.827 0.859 0.812 1.03 55% Snv55 0.611 0.573 0.506 0.95 63% Snv56 0.491 0.501 0.514 0.92 65% Snv57 0.477 0.483 0.542 0.56 53% Snv58 0.612 0.616 0.685 0.62 49% Snv59 0.690 0.625 0.618 0.77 54% Snv61 0.643 0.646 0.681 2.02 75% Snv62 0.578 0.450 0.480 0.65 57% Snv63 0.438 0.463 0.501 0.77 62% Snv64 0.472 0.440 0.450 0.68 60% Snv65 0.509 0.464 0.444 0.74 61% Snv66 0.621 0.608 0.906 0.34 33% Snv67 0.507 0.528 0.641 0.40 42% Snv68 0.425 0.474 0.460 0.38 46% Snv69 0.747 0.712 0.694 0.31 30% Snv70 0.479 0.486 0.478 0.37 43% Snv71 0.471 0.440 0.469 0.35 43% Snv72 0.540 0.542 0.610 0.36 39% Snv73 0.528 0.537 0.535 0.59 53% Snv74 0.464 0.457 0.475 0.13 22% Snv75 0.512 0.490 0.479 0.17 26% Snv76 0.617 0.711 0.735 0.39 36% Snv77 0.554 0.590 0.637 0.15 20% Snv78 0.595 0.730 0.597 0.07 10% Table (5): Method 2 best results (X,Y,Z) (values are in meter) The previous table shows a variation of performance for each satellite from model to model and when using all the Data or a portion of it. Fig (8): Method 2 model performance compared to original offsets Method 2 is slightly different from the first one. The first method forecast future offsets based on past ones while the second one models current errors based on only past broadcast ephemeris. Deep learning techniques prove to be optimal to solve both forecasting and error modeling problems. The main idea in modeling errors using deep learning is to know the initial state and the final state of the phenomenon and the neural network will learn how to solve the problem by knowing the patterns and relations between inputs and outputs (Grossi et al 2008). Therefore, we do not need to know or understand in depth neither the phenomenon, nor the constant or mathematical equations associated with it. The downside of this method is that we don't really know what's going inside the model and we can't convert the neural network into a mathematical problem (Grossi et al 2008).). On the other hand, the fact that the model works shows that these errors can be modelled and that they are not random even if we don't know what's causing them From fig (6) and table (7) we see that the model performance is relied to satellite block. Also, as we see block III has the worst correction percentage of 28%. We believe that there are two reasons: one the lack of Data because they are new satellites and the offsets are already around ~75cm. Blocks IIR and IIRM have the best performance with an improvement of 56% correction percentage. While block IIF is divided to with 57% Satellites which are (57, 62, 60, and 61) and the rest with 41%. Predicting clock bias To predict clock bias () the same model as method 1 was used. With GRU architecture, the next 24h clock biases corrections were predicted by using the past 24h as inputs. The results are represented in table (6) and fig (9). Original 2h 4h 6h 12h 24h Sat_1 14.2 0.31 0.39 0.43 0.65 0.99 Sat_10 15.2 0.51 0.69 0.82 1.26 1.73 Sat_11 18.4 0.45 0.78 1.04 1.71 2.48 Sat_12 14.1 0.40 0.55 0.65 0.91 1.43 Sat_13 14.1 0.58 0.85 1.07 1.56 2.14 Sat_14 15.8 0.31 0.52 0.62 0.97 1.44 Sat_15 14.3 0.37 0.49 0.59 0.88 1.41 Sat_16 14.3 0.43 0.56 0.66 0.91 1.43 Sat_17 14.4 0.82 1.27 1.61 2.35 3.07 Sat_18 16.0 0.32 0.53 0.63 0.95 1.43 Sat_19 14.7 0.37 0.50 0.60 0.93 1.42 Sat_2 14.2 0.38 0.55 0.69 1.06 1.55 Sat_20 14.6 0.47 0.65 0.79 1.14 1.64 Sat_21 14.6 0.48 0.62 0.75 1.07 1.58 Sat_23 15.8 0.30 0.51 0.59 0.94 1.42 Sat_24 14.3 0.29 0.38 0.43 0.67 0.98 Sat_25 14.3 0.28 0.36 0.40 0.61 0.95 Sat_26 14.3 0.31 0.39 0.44 0.67 0.98 Sat_27 14.1 0.32 0.42 0.50 0.75 1.11 Sat_28 15.4 2.01 3.39 4.19 5.17 5.79 Sat_3 13.7 0.67 1.04 1.25 1.68 2.18 Sat_30 14.5 0.30 0.40 0.47 0.72 1.06 Sat_31 14.2 0.47 0.70 0.87 1.31 1.88 Sat_32 14.4 0.27 0.36 0.42 0.64 1.00 Sat_4 15.8 0.29 0.50 0.59 0.96 1.44 Sat_5 14.4 0.53 0.68 0.77 1.03 1.54 Sat_6 14.2 0.29 0.36 0.42 0.62 0.96 Sat_7 14.1 0.53 0.80 1.01 1.47 2.00 Sat_8 14.9 2.19 3.32 3.67 4.37 4.89 Sat_9 14.3 0.28 0.37 0.43 0.64 0.98 mean 14.71 0.52 0.76 0.91 1.29 1.76 Table (6): Clock bias errors model performance through different epochs compared to original data in (Nano-second) The original data has a mean bias of 14.7ns, while the predicted biases have a mean of 0.9ns for 6h in future and 1.7ns for 24h in future. Also, we can see that there are only two satellites with high predicted value (sat28, sat8). Fig (9): Clock bias errors model performance through different epochs compared to original data in (Nano-second) Test in processing This part shows the results of testing our predicted products (method 2 for ephemeris, 6h prediction for clock errors). In real processing problem, the implementation of predicted offsets is quite challenging. However, to keep it simple we replaced ECEF coordinate in Ultra precise ephemeris file by predicted ECEF ephemeris. Also, we replaced a0 in broadcast ephemeris files with predicted ones. Then, we processed three sessions (using RTKlib), one with Broadcast products, the second with precise products and the last one with predicted products. Finally, we tested the results with 4 IGS stations for a random day (4-2-2022). IGS14 coordinates + velocity were used as a reference. For all performed tests, the same processing parameters were used Only GPS satellites data are processed Ionosphere-free combination is used to eliminate the main part of the ionosphere error. Tropospheric zenithal total delays and gradients are estimated DCB correction are included Fig (10): Predicted products performance in GPS processing compared to broadcast and precise products (difference between true coordinate and computed coordinate) Even though the way we have implemented predicted products is not optimal, the result shows an important difference between broadcast and predicted products. However, we want to mention: that, to test our predicted ephemeris we had to remove satellites with large offsets (+25cm) because they make the results noisy and we believe it caused by the interpolation polynomial. Conclusion In this paper, we presented an approach with two methods to reduce broadcast ephemeris and clock errors using deep learning techniques. In order to do that many variations of datasets, architectures, model, layers, and activations function were tried. At the end, the two variables that really matter to get the best performance model are Dataset and model architectures. However, each time a dataset is modified, the performance of each individual satellite changes. Also, it is the same for changing the model architecture; some satellites perform well with the GRU and others with LSTM. Therefore, to achieve the best performance model, a model and dataset optimization must be performed which is difficult and challenging process and not the purpose of this research, the real benefit of this study results are to show that orbits and clocks errors are not random or noise and they could be modeled. Despite of that, these results of ephemerides do not seem practical in GPS case because of ULTRA RAPIDE PRODUCTS. Thus, they may have a use for other constellation (GALILEO, BEIDOU...) and may be used in GPS in some specific applications replacing SBAS Declarations Author Contribution This research is a part of Bousnane. A.A's preparation for PHD, where Berguig. M.C is the first supervisor (for deep learning knowledge) and Dekkiche. H is the second supervisor (for GPS and geodesy knowledge) We, the authors of this manuscript entitled "[ Reducing GPS orbital and clocks errors using recurrent neural networks]," declare the following: The manuscript represents original work that has not been published previously and is not under consideration for publication elsewhere. The authors declare that they have no conflict of interest. This article does not contain any studies with human participants or animals performed by any of the authors. This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Any data, figures, or materials taken from other sources have been properly acknowledged and cited within the manuscript. All GPS Data used in this work are from CDDIS archive: C. Noll, The Crustal Dynamics Data Information System: A resource to support scientific analysis using space geodesy, Advances in Space Research,Volume 45, Issue 12, 15 June 2010, Pages 1421-1440, ISSN 0273-1177, DOI: 10.1016/j.asr.2010.01.018. Acknowledgements: We extend sincere appreciation to Dr. [Berguig. M.C] and [Dekkiche. H] for their invaluable guidance. Special thanks to [Geophysics Laboratory, University of Sciences and Technology Houari Boumediene] for their insights. Lastly, heartfelt appreciation to family and friends for unwavering encouragement. This research was made possible by their collective support and dedication. References Cho, Kyunghyun; van Merrienboer, Bart; Bahdanau, DZmitry; Bougares, Fethi; Schwenk, Holger; Bengio, Yoshua (2014). "Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation". arXiv:1406.1078 CDDIS, Crustal Dynamics Data Information System, https://cddis.nasa.gov/index.html . Google Smartphone Decimeter Challenge, improve high precision GNSS positioning and navigation accuracy on smartphones, 2021, https://www.kaggle.com/competitions/google-smartphone-decimeter-challenge . Google Smartphone Decimeter Challenge, improve high precision GNSS positioning and navigation accuracy on smartphones, 2022, https://www.kaggle.com/competitions/smartphone-decimeter-2022 . Grossi, Enzo & Buscema, Massimo. (2008). Introduction to artificial neural networks. European journal of gastroenterology & hepatology. 19. 1046–54. 10.1097/MEG.0b013e3282f198a0 . He, S., Liu, J., Zhu, X. et al (2023). Research on modeling and predicting of BDS-3 satellite clock bias using the LSTM neural network model. GPS Solut 27, 108. https://doi.org/10.1007/s10291-023-01451-3 International GNSS Service official website, https://igs.org/products/#orbits_clocks . Jason Brownlee (2018), John W. Betz (2016), "Satellite-Based Augmentation Systems," in Engineering Satellite-Based Navigation and Timing: Global Navigation Satellite Systems, Signals, and Receivers, IEEE, pp.201–211, doi: 10.1002/9781119141167.ch8 Munin. E, Blais. A, and Couellan. N (2020), “Convolutional neural network for multipath detection in GNSS receivers,”, doi: 10.1109/AIDA- AT48540.2020.9049188 . NGA GNSS Division Precise Ephemeris Parameters, February 20, 2020, https://earth-info.nga.mil/php/download.php?file=gnss-precise . Orus Perez. R (2019), “Using TensorFlow-based neural network to estimate GNSS single frequency ionospheric delay (IONONet)”, Advances in Space Research, vol. 63, no.5, pp.1607–1618, doi: https://doi.org/10.1016/j.asr.2018.11.011 . Quan. Y (2017), “A new machine learning based method for multi-GNSS data quality assurance and multipath detection,” Ph.D. dissertation, University of Nottingham. Robin M. Schmidt (2019). “Recurrent Neural Networks (RNNs): A gentle Introduction and Overview”. arXiv:1912.05911 Seeber, G. (2003) Satellite Geodesy. 2nd Edition, Walter de Gruyter, Berlin, 589 p. https://doi.org/10.1515/9783110200089 Selbesoglu. M. O (2020), “Prediction of tropospheric wet delay by an artificial neural network model based on meteorological and GNSS data,” Engineering Science and Technology, an International Journal, vol. 23, no. 5, pp. 967–972, doi: https://doi.org/10.1016/j.jestch.2019.11.006 . Sepp Hochreiter, Jürgen Schmidhuber (1997), Long Short-Term Memory. Neural Comput; 9 (8): 1735–1780. doi: https://doi.org/10.1162/neco.1997.9.8.1735 . Sivakrishna. K, Venkata Ratnam. D and Sivavaraprasad. G (2022), "A Bidirectional Deep-Learning Algorithm to Forecast Regional Ionospheric TEC Maps," in IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 15, pp. 4531–4543, doi: 10.1109/JSTARS.2022.3180940 . Srivani. I, Siva Vara Prasad. G and Venkata Ratnam. D (2019), "A Deep Learning-Based Approach to Forecast Ionospheric Delays for GPS Signals," in IEEE Geoscience and Remote Sensing Letters, vol. 16, no. 8, pp. 1180–1184, Aug, doi: 10.1109/LGRS.2019.2895112 . Yamashita, R., Nishio, M., Do, R.K.G. et al (2018). Convolutional neural networks: an overview and application in radiology. Insights Imaging 9, 611–629. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3881623","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":275564688,"identity":"16ca79e1-762a-49d0-bc42-83c1cc7133f7","order_by":0,"name":"Akram Abderrahaman Bousnane","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVRIiWNgGAWjYDACCRiDvYGBgbHhAAPDAaK18ACVHiRNi0QCkVr4Zzcfk/y5wy6PX/KN8eePO+4w8J0noEfizrE0ad4zycWSs3PMJA6eecYgeYCQNTdyzKQZ25gTN9zOMWM42HaYweBgA34d8kAtkj/b6hP33zxj/AGs5TABSwyAWiR42w4nbpDgMZAAazlGQIvhnWPJ1rxtxxNnnEkrkzjb9oxH8gwBLXK3mw/e/NlWndjffnjzh8q2O3IEQwwD8JCofhSMglEwCkYBNgAAMHdOTzhRcxcAAAAASUVORK5CYII=","orcid":"","institution":"Geophysics Laboratory, University of Sciences and Technology Houari Boumediene","correspondingAuthor":true,"prefix":"","firstName":"Akram","middleName":"Abderrahaman","lastName":"Bousnane","suffix":""},{"id":275564689,"identity":"b37a5d0b-89d4-48ab-aa5e-1a69cfa4bb47","order_by":1,"name":"Mohamed Cherif Berguig","email":"","orcid":"","institution":"Geophysics Laboratory, University of Sciences and Technology Houari 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\u003cstrong\u003e(15).\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"Fig1.png","url":"https://assets-eu.researchsquare.com/files/rs-3881623/v1/53a5627154c6b2fc99d77fdd.png"},{"id":51948132,"identity":"e9d5a5ff-6b6d-44a5-8ac9-762ca0773eec","added_by":"auto","created_at":"2024-03-04 11:55:58","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":107629,"visible":true,"origin":"","legend":"\u003cp\u003eStructure of LSTM cell \u003cstrong\u003e(\u003c/strong\u003eSepp Hochreiter 1997\u003cstrong\u003e).\u003c/strong\u003e\u003c/p\u003e","description":"","filename":"fig2.png","url":"https://assets-eu.researchsquare.com/files/rs-3881623/v1/90dd357ae70ebe0c20be3420.png"},{"id":51948455,"identity":"66637d75-30fd-485a-8fe9-72f337eefb28","added_by":"auto","created_at":"2024-03-04 12:03:58","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":19801,"visible":true,"origin":"","legend":"\u003cp\u003eStructure of GRU cell (Cho et al 2014)\u003c/p\u003e","description":"","filename":"fig3.png","url":"https://assets-eu.researchsquare.com/files/rs-3881623/v1/5b2c1f20a82e9a36365d6bb6.png"},{"id":51948128,"identity":"67454914-9170-48c0-89e8-c1a702c127fc","added_by":"auto","created_at":"2024-03-04 11:55:58","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":68545,"visible":true,"origin":"","legend":"\u003cp\u003eProposed model for this study.\u003c/p\u003e","description":"","filename":"Fig4.png","url":"https://assets-eu.researchsquare.com/files/rs-3881623/v1/7c1391a039faf373080c5fbc.png"},{"id":51948134,"identity":"cc4dc9a7-7f92-4580-91b7-b17af7bdeec2","added_by":"auto","created_at":"2024-03-04 11:55:58","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":81089,"visible":true,"origin":"","legend":"\u003cp\u003eMethod 1 diagram with one input and one output\u003c/p\u003e","description":"","filename":"Fig5.png","url":"https://assets-eu.researchsquare.com/files/rs-3881623/v1/b5765a1fc0d3514e7cde9308.png"},{"id":51948130,"identity":"9369b25a-5ecd-4ce8-91db-f60ba8eec602","added_by":"auto","created_at":"2024-03-04 11:55:58","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":27288,"visible":true,"origin":"","legend":"\u003cp\u003eEphemeris model 1 performance through different epochs compared to original data\u003c/p\u003e","description":"","filename":"Fig6.png","url":"https://assets-eu.researchsquare.com/files/rs-3881623/v1/bd5f5c7cd8ea97eaddace46f.png"},{"id":51948137,"identity":"3bc394b0-de77-487f-a5d4-890adb87fdf1","added_by":"auto","created_at":"2024-03-04 11:55:58","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":29977,"visible":true,"origin":"","legend":"\u003cp\u003eMethod 2 diagram with one input and one output\u003c/p\u003e","description":"","filename":"Fig7.png","url":"https://assets-eu.researchsquare.com/files/rs-3881623/v1/490962c62d56b82bfe7f8f47.png"},{"id":51948136,"identity":"2c144d69-6dd0-4e0d-8dce-0cb127bd0227","added_by":"auto","created_at":"2024-03-04 11:55:58","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":21433,"visible":true,"origin":"","legend":"\u003cp\u003eMethod 2 model performance compared to original offsets\u003c/p\u003e","description":"","filename":"Fig8.png","url":"https://assets-eu.researchsquare.com/files/rs-3881623/v1/54f68ec52e05d48aa72d2631.png"},{"id":51948135,"identity":"944b23ee-6a2f-469a-b75b-eef83a2df485","added_by":"auto","created_at":"2024-03-04 11:55:58","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":25653,"visible":true,"origin":"","legend":"\u003cp\u003eClock bias errors model performance through different epochs compared to original data in (Nano-second)\u003c/p\u003e","description":"","filename":"Fig9.png","url":"https://assets-eu.researchsquare.com/files/rs-3881623/v1/6d2ea70917fd0c5082fb3df6.png"},{"id":51948131,"identity":"98d504c8-26ff-4062-927a-d1fa5d116e77","added_by":"auto","created_at":"2024-03-04 11:55:58","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":284152,"visible":true,"origin":"","legend":"\u003cp\u003ePredicted products performance in GPS processing compared to broadcast and precise products (difference between true coordinate and computed coordinate)\u003c/p\u003e","description":"","filename":"Fig10.png","url":"https://assets-eu.researchsquare.com/files/rs-3881623/v1/744df648a04d78adaef1b0c3.png"},{"id":52701321,"identity":"18ec8ccf-8574-43f4-ba13-4a39266fccb8","added_by":"auto","created_at":"2024-03-14 17:29:54","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1145480,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3881623/v1/2b2d661e-2e75-4d93-8a77-90226b87561d.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Reducing GPS orbital and clocks errors using recurrent neural networks","fulltext":[{"header":"Introduction","content":"\u003cp\u003eGNSS satellites fly in very precise, well-known trajectory (orbit), defined by Keplerian ellipse. However, the gravitational and non-gravitational forces deviate the satellite from its original orbit. Therefore, ground control system continually monitors this orbits changes and provide us with what we call broadcast ephemeris transmitted in navigation message in addition to clocks bias and drift. Even with the correction done by the ground control system. Small errors still exist that can result around ~\u0026thinsp;1.0 meter in position (IGS official website) and 5 ns in clock (IGS official website).\u003c/p\u003e \u003cp\u003eFor some applications, these last-mentioned errors could be considered as significant. The common ways to reduce them is to use precise products provided by GNSS ground control services, like the International GNSS Service (IGS) products. IGS provides three types of products: Ultra rapid, rapid, and final products. The three products are sampled on 15mn time interval with a precision of 0.05m \u0026ndash; 0.025m for orbits and ~\u0026thinsp;3ns \u0026minus;\u0026thinsp;75ps for clocks. Another way to compensate those errors in single point positioning mode, is by using SBAS system or Precise Point Positioning (John W et al 2016).\u003c/p\u003e \u003cp\u003eThis study proposes another way to reduce broadcast ephemeris and clock bias errors by using deep learning techniques. Deep learning (DL) is a very powerful technique in handling time-series data, and it can learn time dependent patterns across multiple models \u003cb\u003e(\u003c/b\u003eGrossi et al 2008\u003cb\u003e)\u003c/b\u003e. Therefore, DL can be found useful in the discovery of unknown and hidden information in GNSS data. The large quantity of data provided by IGS, made the development of a Dl model possible.\u003c/p\u003e \u003cp\u003eThe proposed methodology treats the problem in two different methods: one by creating a model that forecasts the future ephemeris and clock offsets based on the past ones. The other is to create a deep learning model that predicts the current orbit errors based on past orbital elements (Keplerian elements).\u003c/p\u003e \u003cp\u003eUp to day several research have implemented machine learning algorithms in GNSS field, (Srivani. I et al 2019) (Sivakrishna. K et al 2022) (Orus Perez. R 2019) and others treated ionosphere errors in different ways using deep learning. An ANN (Artificial Neural Network) model was implemented to predict tropospheric wet delay based on meteorological and GNSS data (Selbesoglu. M.O. 2020). Different machine learning techniques were used to detect multipath (Munin. E et al 2020) (Quan. Y 2017). Also, Google in collaboration with the Institute of Navigation (ION) lunched two competitions for 10.000\u003cspan\u003e$\u003c/span\u003e prize pool on Kaggle to compute smartphone\u0026rsquo;s location down to the decimeter or even centimeter resolution using ML (Google Smartphone Decimeter Challenge \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) (Google Smartphone Decimeter Challenge \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Recently (He, S 2023) used LSTM to model and predict BDS-3 satellite clock bias.\u003c/p\u003e \u003cp\u003eDeep learning is a subfield of machine learning that involves training artificial neural networks with multiple layers. These neural networks are inspired by the structure and function of the human brain and are able to learn and improve their performance over time through a process known as training (Grossi et al 2008). Because we have non-linear and multi-Dimension datasets, the best suitable ML technique in this case is deep learning (Jason Brownlee \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). There are several deep learning architectures that are particularly well suited for time series forecasting such as: Recurrent Neural Networks (RNNs) with its popular variation Long Short-Term Memory (Sepp Hochreiter et al 1997) and (Gated Recurrent Unite) (Cho et al \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2014\u003c/span\u003e), and Convolutional Neural Networks (Convolutional neural networks) (Yamashita et al \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). In this study the three architectures mentioned before are used. However, only LSTM and GRU results will be shared in this paper.\u003c/p\u003e"},{"header":"Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eRecurrent neural networks\u003c/h2\u003e \u003cp\u003eArtificial Neural Networks algorithms are a simulation of human brain behavior. This means ANN algorithms learn patterns from massive historical or past data, and apply gained knowledge to predict new results (Grossi et al 2008).\u003c/p\u003e \u003cp\u003eRecurrent Neural Networks (RNNs) are a type of artificial neural network that can handle sequential data by allowing information to be passed from one time step to the next, when simple ANN view each observation as independent because the network is not able to retain time steps informations \u003cb\u003e(\u003c/b\u003eRobin M. Schmidt \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2019\u003c/span\u003e\u003cb\u003e).\u003c/b\u003e\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eLet's consider a simple RNN with one input, one hidden layer, and one output (Fig.\u0026nbsp;1). At each time step t, the RNN takes an input vector x(t) and produces an output vector y(t) and a(t).\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;(1) : Structure of simple RNN cell \u003cb\u003e(15).\u003c/b\u003e\u003c/p\u003e \u003cp\u003eThe hidden layer at time step t is denoted by h(t), which is a function of the previous hidden state a(t-1) and the current input x(t).\u003c/p\u003e \u003cp\u003ea(t)\u0026thinsp;=\u0026thinsp;f(Wx(t)\u0026thinsp;+\u0026thinsp;Wa(t-1)\u0026thinsp;+\u0026thinsp;Ba)\u003c/p\u003e \u003cp\u003ey(t)\u0026thinsp;=\u0026thinsp;g(Wy(t)\u0026thinsp;+\u0026thinsp;By)\u003c/p\u003e \u003cp\u003ewhere f and g are non-linear activation function (tanh, sigmoid, \u0026hellip;.), Wx, Wa, Wy are weights for the recurrent, input connections, and output, respectively, and Ba, By are bias vectors.\u003c/p\u003e \u003cp\u003eCombining these blocks will give us an RNN model shown below (Fig.\u0026nbsp;4). To train an RNN, we need to compute the loss at each time step and then back-propagate the errors through time. The loss function is typically defined as the sum of the losses at each time step:\u003c/p\u003e \u003cp\u003eL = \u0026sum; L(t)\u003c/p\u003e \u003cp\u003ewhere L(t) is the loss for one time step t. We can then compute the gradient of the loss with respect to the parameters of the RNN using the Back-Propagation Through Time (BPTT) algorithm.\u003c/p\u003e \u003cp\u003eThe BPTT algorithm works by unrolling the RNN through time and computing the gradients at each time step. The gradients are then accumulated over time and used to update the weights and biases using gradient descent a similar optimization algorithm.\u003c/p\u003e \u003cp\u003e However, the problem with RNN is exploding/vanishing gradients during back-propagation step. Because of these problems, RNN cannot capture the relevant information from long-term dependencies because of multiplicative gradient values that can gradually increase/decrease based on the number of hidden layers.\u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;(2): Structure of LSTM cell \u003cb\u003e(\u003c/b\u003eSepp Hochreiter \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e1997\u003c/span\u003e\u003cb\u003e).\u003c/b\u003e\u003c/p\u003e \u003cp\u003eLong-Short-Term-Memory \u003cb\u003e(\u003c/b\u003eSepp Hochreiter et al 1997\u003cb\u003e)\u003c/b\u003e LSTM and GRU (Cho et al \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) are a special type of RNNs mainly developed to tackle the main problem of simple RNNs, the problem of vanishing gradients. A common LSTM unit is composed of a cell (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({C}_{t}\\)\u003c/span\u003e\u003c/span\u003e), an input gate \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(({i}_{t}\\)\u003c/span\u003e\u003c/span\u003e), an output gate (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({o}_{t}\\)\u003c/span\u003e\u003c/span\u003e) and a forget gate \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({(f}_{t}\\)\u003c/span\u003e\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe cell state allows information to flow along the entire chain with only minor linear actions through three gates. The three gates work as filters and control the flow of information and determine which information is kept or disregarded. Forget gates decide what information to discard from a previous state (long-term memory). The value of 1 means keeps the information and the value of 0 means to discard it. Input gates decide which pieces of new information to store in the current state. Output gates control which pieces of information in the current state to output (short-term memory). Both input and output gates use the same system as forget gate bounding values between 0 and 1 by sigmoid.\u003c/p\u003e \u003cp\u003eThe forms of the equations for an LSTM with a forget gate are (Sepp Hochreiter et al 1997)\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({f}_{t}= \\sigma ({W}_{f,x}{x}_{t}+{W}_{f,h}{h}_{t-1}+{b}_{f}\\)\u003c/span\u003e \u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({i}_{t}= \\sigma ({W}_{i,x}{x}_{t}+{W}_{i,h}{h}_{t-1}+{b}_{i})\\)\u003c/span\u003e \u003c/span\u003e \u003cdiv id=\"Equb\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$${o}_{t}= \\sigma ({W}_{o,x}{x}_{t}+{W}_{o,h}{h}_{t-1}+{b}_{o})$$\u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Equc\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$${\\stackrel{\\sim}{C}}_{t}= tanh({W}_{c,x}{x}_{t}+{W}_{c,h}{h}_{t-1}+{b}_{c})$$\u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Equd\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$${C}_{t}= {f}_{t}*{C}_{t-1}+{i}_{t}*{\\stackrel{\\sim}{C}}_{t}$$\u003c/div\u003e \u003c/div\u003e \u003cdiv id=\"Eque\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$${h}_{t}= {o}_{t}*\\text{t}\\text{a}\\text{n}\\text{h}\\left({C}_{t}\\right)$$\u003c/div\u003e \u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;(3) : Structure of GRU cell (Cho et al \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2014\u003c/span\u003e)\u003c/p\u003e \u003cp\u003eGRU is a variant of the LSTM which instead of the input, forget, and output gates in LSTM; GRU has two gates: an update gate z, and a reset gate r.\u003c/p\u003e \u003cp\u003eThe reset gate (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({r}_{t}\\)\u003c/span\u003e\u003c/span\u003e) is responsible for the short-term memory as it decides how much past information is kept and disregarded. Otherwise, the update gate (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({z}_{t}\\)\u003c/span\u003e\u003c/span\u003e) is responsible for the long-term memory and is comparable to the LSTM\u0026rsquo;s forget gate.\u003c/p\u003e \u003cp\u003eGRU have variant forms. The common form of GRU is fully gated unit (Fig.\u0026nbsp;3.) and have the equations \u003cb\u003e(\u003c/b\u003eCho et al \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2014\u003c/span\u003e):\u003cdiv id=\"Equf\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equf\" name=\"EquationSource\"\u003e\n$${r}_{t}= \\sigma ({W}_{r,x}{x}_{t}+{W}_{r,h}{h}_{t-1}+{b}_{r})$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equg\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equg\" name=\"EquationSource\"\u003e\n$${z}_{t}= \\sigma ({W}_{z,x}{x}_{t}+{W}_{z,h}{h}_{t-1}+{b}_{z})$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equh\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equh\" name=\"EquationSource\"\u003e\n$${C}_{t}= tanh({W}_{c,x}{x}_{t}+{W}_{c,h}({r}_{t}*{h}_{t-1})+{b}_{c})$$\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equi\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equi\" name=\"EquationSource\"\u003e\n$${h}_{t}= {z}_{t}*{h}_{t-1}+\\left(1-{z}_{t}\\right)\\text{*}{C}_{t}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eDue to the simpler architecture compared to LSTM, GRUs are computationally more efficient and faster to train as they need less memory. However, as GRUs do not have a separate hidden and cell state gate, they might not be able to consider observations as far into the past as the LSTM.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure\u0026nbsp;(4): Proposed model for this study.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003eData description:\u003c/h2\u003e \u003cp\u003eFor this work, two different methods were applied. In which the output for both methods are quite similar (broadcast ephemeris corrections) while the input is different. For method 1 inputs are differences between broadcast and precise ephemeris, while the inputs in second methods are Keplerian element from broadcast ephemeris only.\u003c/p\u003e \u003cp\u003eTo create our deep learning models, broadcast ephemeris and precise ephemeris were used for all the operational satellites from the lunch date until December 2021.\u003c/p\u003e \u003cp\u003eAll the files were downloaded from the CDDIS service database (Crustal Dynamics Data Information System Database). Then the ECEF broadcast coordinate was calculated using the algorithm given by (Seeber \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2003\u003c/span\u003e) An interval sample of one hour was used.\u003c/p\u003e \u003cp\u003eFor each satellite the orbital elements \u003cem\u003e{M, w, e, cuc, cus, crc, crs, Ɵ, i}\u003c/em\u003e were taken. For precise ephemeris, the coordinate for each same epoch (1h) was taken.\u003c/p\u003e \u003cp\u003eThe precise ephemeris products provide the center-of-mass (CoM) coordinates of the respective satellite, while the broadcast ephemeris represents the satellite antenna phase center coordinates. In order to compare the two products, antennas offset correction (AOC) is needed for precise ephemeris. Thus, we used antenna parameters published by National Geospatial-Intelligence Agency (NGA) (NGA 2020).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eOffsets between precise ephemeris and broadcast ephemeris, mean from 1-1-2022 to 15-08-2022 (Test Data)\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSNV\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cem\u003eDx\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cem\u003eDy\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cem\u003eDz\u003c/em\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv43\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN13\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.915\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.790\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.886\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv45\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN21\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.711\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.712\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e3.164\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv48\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN7\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.516\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.710\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.701\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv50\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN5\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.972\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.776\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.899\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv51\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN20\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.856\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.930\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.220\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv52\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN31\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.220\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.935\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.226\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv53\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN17\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.745\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.941\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.892\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv55\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN15\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.600\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.370\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.560\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv56\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN16\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.526\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.059\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.671\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv57\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN29\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.821\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.053\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.312\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv58\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN12\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.804\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.580\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.397\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv59\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN19\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.561\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.223\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.454\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv61\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN2\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.987\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e2.907\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e3.128\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv62\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN25\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.110\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.102\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.259\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv63\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN1\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.220\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.192\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.290\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv64\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN30\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.080\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.992\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.321\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv65\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN24\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.196\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.165\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.291\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv66\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN27\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.082\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.015\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.072\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv67\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN6\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.959\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.806\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.108\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv68\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN9\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.672\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.841\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv69\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN3\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.904\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.106\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.067\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv70\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN32\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.762\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.926\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.860\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv71\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN26\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.891\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.739\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.804\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv72\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN8\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.864\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.920\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv73\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN10\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.936\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.177\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.259\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv74\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN4\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.585\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.586\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.628\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv75\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN18\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.695\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.636\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.661\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv76\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN23\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.116\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e1.072\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.054\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv77\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN14\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.628\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.771\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.834\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eSnv78\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003ePRN11\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.628\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.761\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.740\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;(1) shows the offsets of the data that we used to test our models. The average of those offsets is 1.2m. However, we can see that two satellites have large offsets (SNV45, SNV61) both of them are in IIR block.\u003c/p\u003e \u003c/div\u003e"},{"header":"Results and discussion","content":"\u003cp\u003e\u003cstrong\u003eMethod 1:\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe first method is to predict the future offsets based on the past ones. Therefore, we used the last 24h data (both broadcast and precise ephemeris) to predict the next 72h offsets (Many to one model).\u003c/p\u003e\n\u003cp\u003eFig (5): Method 1 diagram with one input and one output\u003c/p\u003e\n\u003cp\u003eAs in fig (5) we can see that, the input victor is a 3D matrix with the shape (\u003cstrong\u003e\u003cem\u003en\u003c/em\u003e\u003c/strong\u003e, 3, 24); where \u003cstrong\u003e\u003cem\u003en\u0026nbsp;\u003c/em\u003e\u003c/strong\u003eis the length of the dataset, 3 refers to the three ECEF offset component (Dx, Dy, Dz), and 24 refers to the last 24 epoch. \u0026nbsp;The output is a 2D victor (\u003cstrong\u003e\u003cem\u003en\u003c/em\u003e\u003c/strong\u003e, 3).\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\" width=\"348\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e1h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e2h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e6h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e12h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e24h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e48h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e72h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv43\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.24\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.29\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.44\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.47\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv45\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.51\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.62\u003c/p\u003e\n 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valign=\"top\"\u003e\n \u003cp\u003e0.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.38\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.44\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv51\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.49\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.49\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv52\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.21\u003c/p\u003e\n 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valign=\"top\"\u003e\n \u003cp\u003e0.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.49\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv56\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.43\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv57\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.15\u003c/p\u003e\n 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valign=\"top\"\u003e\n \u003cp\u003e0.18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.34\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.41\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.45\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv61\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.21\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.44\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.55\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.68\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.81\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv62\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.10\u003c/p\u003e\n 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valign=\"top\"\u003e\n \u003cp\u003e0.11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.42\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.41\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv65\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.43\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv66\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n 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valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv68\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.29\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.40\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv69\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.11\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.42\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.54\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.56\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n 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\u003cp\u003e0.54\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv73\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.47\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.50\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv74\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.24\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.59\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv75\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.23\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.41\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.55\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.62\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv76\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.47\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.53\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.62\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.78\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.94\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv77\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.29\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.42\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.78\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.697406340057636%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv78\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.832853025936599%\" valign=\"top\"\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.680115273775217%\" valign=\"top\"\u003e\n \u003cp\u003e0.38\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.39193083573487%\" valign=\"top\"\u003e\n \u003cp\u003e0.44\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.54\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.527377521613833%\" valign=\"top\"\u003e\n \u003cp\u003e0.62\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.815561959654179%\" valign=\"top\"\u003e\n \u003cp\u003e0.71\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eTable 2: Method 1 mean (Dx, Dy, Dz) results over different epochs\u003c/p\u003e\n\u003cp\u003eFig (6): Ephemeris model 1 performance through different epochs compared to original data\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\" width=\"696\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"28.448275862068964%\" valign=\"top\"\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd width=\"9.626436781609195%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e1h-2h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.620689655172415%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e2h-4h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.626436781609195%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e4h-6h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.057471264367816%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e6h-12h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e12h-24h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e24h-48h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e48h-72h\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"28.448275862068964%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eDecay over epochs\u0026nbsp;\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.626436781609195%\" valign=\"top\"\u003e\n \u003cp\u003e0.062\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.620689655172415%\" valign=\"top\"\u003e\n \u003cp\u003e0.078\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.626436781609195%\" valign=\"top\"\u003e\n \u003cp\u003e0.048\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.057471264367816%\" valign=\"top\"\u003e\n \u003cp\u003e0.059\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e0.090\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e0.060\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e0.050\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"28.448275862068964%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eDecay percent\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.626436781609195%\" valign=\"top\"\u003e\n \u003cp\u003e41.98%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.620689655172415%\" valign=\"top\"\u003e\n \u003cp\u003e37.20%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.626436781609195%\" valign=\"top\"\u003e\n \u003cp\u003e16.62%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.057471264367816%\" valign=\"top\"\u003e\n \u003cp\u003e17.67%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e22.93%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e12.48%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e9.12%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"28.448275862068964%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eDecay per hour\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.626436781609195%\" valign=\"top\"\u003e\n \u003cp\u003e0.062\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.620689655172415%\" valign=\"top\"\u003e\n \u003cp\u003e0.039\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.626436781609195%\" valign=\"top\"\u003e\n \u003cp\u003e0.024\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.057471264367816%\" valign=\"top\"\u003e\n \u003cp\u003e0.010\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e0.008\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e0.003\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e0.002\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"28.448275862068964%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eDecay percent per hour\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.626436781609195%\" valign=\"top\"\u003e\n \u003cp\u003e41.98%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"8.620689655172415%\" valign=\"top\"\u003e\n \u003cp\u003e18.60%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"9.626436781609195%\" valign=\"top\"\u003e\n \u003cp\u003e8.31%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"10.057471264367816%\" valign=\"top\"\u003e\n \u003cp\u003e2.95%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e1.91%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e0.52%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.206896551724139%\" valign=\"top\"\u003e\n \u003cp\u003e0.38%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eTable (3): Models performance decay over time\u003c/p\u003e\n\u003cp\u003eFrom the previous results, Fig (6) and tables (2 and 3), we can see that the original distances are around (1.00-1.50m) except for the new satellites (74-78) (block III) which have distances below 0.90m. The problem with the new satellites is the lack of data which explains the underperformance of the model for these satellites. Also, satellites (61,45) which have an original error of ~2.67m, the model has the worst performance.\u003c/p\u003e\n\u003cp\u003eWithout the satellites (i.e., 45, 61, 74-78), an offset of (\u0026gt; 28 cm) will be obtained at 12h and (~ 50cm) at 72h. Taking into the account that, this model is a single model which process all the satellites using GRU architecture. In addition, we may notice that the rate of degradation of model performance gets lower over time.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMethod 2\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eFig (7): Method 2 diagram with one input and one output\u003c/p\u003e\n\u003cp\u003eThe second method is an error reduction for broadcast ephemeris. \u0026nbsp;The inputs in this method are Keplerian elements of the last 12 sets of ephemeris (24h) and the output is the correction for the current satellite coordinates (Dx, Dy, Dz). Therefore, no precise ephemeris are required in this model. They are only needed for the training.\u003c/p\u003e\n\u003cp\u003eAlso, this method is less accurate than the first, but it turned out to be more practical.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMethod 2 results\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\" width=\"597\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eLSTM\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eGRU\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eBILSTM\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eBIGRU\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eLSTM (2009 Data)\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv43\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.497\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.515\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.532\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.510\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.531\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv45\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.687\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.735\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.784\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.709\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.659\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv48\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.516\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.516\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.556\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.536\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.720\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv50\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.544\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.565\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.580\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.536\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.544\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv51\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.833\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.757\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.873\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.810\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.902\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv52\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.449\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.473\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.472\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.443\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.491\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv53\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.832\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.888\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.893\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.887\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.833\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv55\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.625\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.671\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.563\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.635\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.587\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv56\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.502\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.561\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.562\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.556\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.741\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv57\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.501\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.556\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.524\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.514\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.640\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv58\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.645\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.639\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.637\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.642\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.726\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv59\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.725\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.753\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.644\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.674\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.709\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv61\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.786\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.698\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.849\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.753\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.657\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv62\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.561\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.540\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.524\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.533\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.503\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv63\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.498\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.524\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.479\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.485\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.467\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv64\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.460\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.463\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.473\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.454\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.457\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv65\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.517\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.522\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.475\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.473\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.517\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv66\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.905\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.712\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.966\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.795\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e1.036\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv67\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.562\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.601\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.575\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.558\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.651\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv68\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.453\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.468\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.476\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.470\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.466\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv69\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.739\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.729\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.723\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.717\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.733\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv70\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.504\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.492\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.509\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.495\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.481\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv71\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.460\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.467\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.466\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.492\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.566\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv72\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.564\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.597\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.569\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.572\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.605\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv73\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.533\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.594\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.578\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.561\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.567\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv74\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.479\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.488\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.475\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.465\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.487\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv75\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.504\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.529\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.494\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.519\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.520\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv76\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.744\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.770\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.737\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.767\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.688\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv77\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.594\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.615\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.602\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.598\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.610\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"16.778523489932887%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv78\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.107382550335572%\" valign=\"top\"\u003e\n \u003cp\u003e0.648\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"15.604026845637584%\" valign=\"top\"\u003e\n \u003cp\u003e0.691\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"18.456375838926174%\" valign=\"top\"\u003e\n \u003cp\u003e0.647\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.610738255033556%\" valign=\"top\"\u003e\n \u003cp\u003e0.693\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.44295302013423%\" valign=\"top\"\u003e\n \u003cp\u003e0.640\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eTable (4): Method 2 results for different RNN architectures\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\" width=\"605\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eDx\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eDy\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eDz\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eCorrection\u0026nbsp;\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eCorrection Percentage\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003e\u0026nbsp;\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv43\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.503\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.498\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.490\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e42%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv45\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.653\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.697\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.628\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e2.20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e77%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv48\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.498\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.528\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.521\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e1.13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e69%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv50\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.510\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.544\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.553\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e39%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv51\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.722\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.757\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.794\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.24\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e24%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv52\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.459\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.433\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.436\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.68\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e61%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv53\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.827\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.859\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.812\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e1.03\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e55%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv55\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.611\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.573\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.506\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.95\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e63%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv56\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.491\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.501\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.514\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.92\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e65%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv57\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.477\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.483\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.542\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e53%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv58\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.612\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.616\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.685\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.62\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e49%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv59\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.690\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.625\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.618\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.77\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e54%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv61\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.643\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.646\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.681\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e2.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e75%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv62\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.578\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.450\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.480\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.65\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e57%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv63\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.438\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.463\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.501\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.77\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e62%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv64\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.472\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.440\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.450\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.68\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e60%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv65\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.509\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.464\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.444\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.74\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e61%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv66\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.621\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.608\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.906\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.34\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e33%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv67\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.507\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.528\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.641\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e42%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv68\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.425\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.474\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.460\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.38\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e46%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv69\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.747\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.712\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.694\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e30%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv70\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.479\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.486\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.478\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e43%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv71\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.471\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.440\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.469\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e43%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv72\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.540\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.542\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.610\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e39%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv73\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.528\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.537\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.535\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.59\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e53%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv74\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.464\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.457\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.475\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e22%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv75\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.512\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.490\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.479\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e26%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv76\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.617\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.711\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.735\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e36%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv77\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.554\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.590\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.637\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e20%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"15.867768595041323%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cstrong\u003e\u003cem\u003eSnv78\u003c/em\u003e\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.066115702479339%\" valign=\"top\"\u003e\n \u003cp\u003e0.595\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.892561983471074%\" valign=\"top\"\u003e\n \u003cp\u003e0.730\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.900826446280991%\" valign=\"top\"\u003e\n \u003cp\u003e0.597\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"20.99173553719008%\" valign=\"top\"\u003e\n \u003cp\u003e0.07\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"26.28099173553719%\" valign=\"top\"\u003e\n \u003cp\u003e10%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eTable (5): Method 2 best results (X,Y,Z) (values are in meter)\u003c/p\u003e\n\u003cp\u003eThe previous table shows a variation of performance for each satellite from model to model and when using all the Data or a portion of it.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFig (8): Method 2 model performance compared to original offsets\u003c/p\u003e\n\u003cp\u003eMethod 2 is slightly different from the first one. The first method forecast future offsets based on past ones while the second one models current errors based on only past broadcast ephemeris. Deep learning techniques prove to be optimal to solve both forecasting and error modeling problems.\u003c/p\u003e\n\u003cp\u003eThe main idea in modeling errors using deep learning is to know the initial state and the final state of the phenomenon and the neural network will learn how to solve the problem by knowing the patterns and relations between inputs and outputs (Grossi et al 2008). \u0026nbsp;Therefore, we do not need to know or understand in depth neither the phenomenon, nor the constant or mathematical equations associated with it. \u0026nbsp; The downside of this method is that we don\u0026apos;t really know what\u0026apos;s going inside the model and we can\u0026apos;t convert the neural network into a mathematical problem (Grossi et al 2008).). \u0026nbsp;\u003c/p\u003e\n\u003cp\u003eOn the other hand, the fact that the model works shows that these errors can be modelled and that they are not random even if we don\u0026apos;t know what\u0026apos;s causing them\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFrom fig (6) and table (7) we see that the model performance is relied to satellite block. Also, as we see block III has the worst correction percentage of 28%. We believe that there are two reasons: one the lack of Data because they are new satellites and the offsets are already around ~75cm. Blocks IIR and IIRM have the best performance with an improvement of 56% correction percentage. While block IIF is divided to with 57% Satellites which are (57, 62, 60, and 61) and the rest with 41%.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003ePredicting clock bias\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eTo predict clock bias () the same model as method 1 was used. With GRU architecture, the next 24h clock biases corrections were predicted by using the past 24h as inputs. The results are represented in table (6) and fig (9).\u0026nbsp;\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\" width=\"423\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\u003cbr\u003e\u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eOriginal\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003e2h\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003e4h\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003e6h\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003e12h\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003e24h\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_1\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e0.65\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e0.99\u003c/p\u003e\n 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valign=\"top\"\u003e\n \u003cp\u003e14.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.55\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.65\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e0.91\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e1.43\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_13\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n 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valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_19\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e0.93\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e1.42\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_2\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.38\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.55\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.69\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e1.06\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e1.55\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_20\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.47\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.65\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.79\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e1.14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e1.64\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_21\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.62\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e1.07\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e1.58\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_23\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e15.8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.51\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.59\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e0.94\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e1.42\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_24\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.29\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.38\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e0.98\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_25\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e0.61\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e0.95\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_26\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.44\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e0.98\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_27\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.42\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e0.75\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e1.11\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_28\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e15.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e2.01\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e3.39\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e4.19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e5.17\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e5.79\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_3\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e13.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.67\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e1.04\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e1.25\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e1.68\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e2.18\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_30\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.47\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e0.72\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e1.06\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_31\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.47\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.70\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.87\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e1.31\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e1.88\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_32\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.36\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n 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valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_7\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.53\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.80\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e1.01\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e1.47\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e2.00\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_8\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e2.19\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e3.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e3.67\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e4.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e4.89\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003eSat_9\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e0.64\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e0.98\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"17.494089834515368%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003emean\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"19.38534278959811%\" valign=\"top\"\u003e\n \u003cp\u003e14.71\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.42080378250591%\" valign=\"top\"\u003e\n \u003cp\u003e0.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.056737588652481%\" valign=\"top\"\u003e\n \u003cp\u003e0.76\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"13.002364066193854%\" valign=\"top\"\u003e\n \u003cp\u003e0.91\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"11.347517730496454%\" valign=\"top\"\u003e\n \u003cp\u003e1.29\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"12.293144208037825%\" valign=\"top\"\u003e\n \u003cp\u003e1.76\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eTable (6): Clock bias errors model performance through different epochs compared to original data in (Nano-second)\u003c/p\u003e\n\u003cp\u003eThe original data has a mean bias of 14.7ns, while the predicted biases have a mean of 0.9ns for 6h in future and 1.7ns for 24h in future. Also, we can see that there are only two satellites with high predicted value (sat28, sat8).\u003c/p\u003e\n\u003cp\u003eFig (9): Clock bias errors model performance through different epochs compared to original data in (Nano-second)\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTest in processing\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis part shows the results of testing our predicted products (method 2 for ephemeris, 6h prediction for clock errors). In real processing problem, the implementation of predicted offsets is quite challenging. However, to keep it simple we replaced ECEF coordinate in Ultra precise ephemeris file by predicted ECEF ephemeris. Also, we replaced a0 in broadcast ephemeris files with predicted ones. Then, we processed three sessions (using RTKlib), one with Broadcast products, the second with precise products and the last one with predicted products. Finally, we tested the results with 4 IGS stations for a random day (4-2-2022). IGS14 coordinates + velocity were used as a reference.\u003c/p\u003e\n\u003cp\u003eFor all performed tests, the same processing parameters were used\u0026nbsp;\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003eOnly GPS satellites data are processed \u0026nbsp;\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eIonosphere-free combination is used to eliminate the main part of the ionosphere error.\u003c/li\u003e\n \u003cli\u003eTropospheric zenithal total delays and gradients are estimated\u003c/li\u003e\n \u003cli\u003eDCB correction are included\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eFig (10): Predicted products performance in GPS processing compared to broadcast and precise products (difference between true coordinate and computed coordinate)\u003c/p\u003e\n\u003cp\u003eEven though the way we have implemented predicted products is not optimal, the result shows an important difference between broadcast and predicted products. However, we want to mention: that, to test our predicted ephemeris we had to remove satellites with large offsets (+25cm) because they make the results noisy and we believe it caused by the interpolation polynomial.\u0026nbsp;\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eIn this paper, we presented an approach with two methods to reduce broadcast ephemeris and clock errors using deep learning techniques. In order to do that many variations of datasets, architectures, model, layers, and activations function were tried. At the end, the two variables that really matter to get the best performance model are Dataset and model architectures. However, each time a dataset is modified, the performance of each individual satellite changes. Also, it is the same for changing the model architecture; some satellites perform well with the GRU and others with LSTM. Therefore, to achieve the best performance model, a model and dataset optimization must be performed which is difficult and challenging process and not the purpose of this research, the real benefit of this study results are to show that orbits and clocks errors are not random or noise and they could be modeled.\u003c/p\u003e \u003cp\u003eDespite of that, these results of ephemerides do not seem practical in GPS case because of ULTRA RAPIDE PRODUCTS. Thus, they may have a use for other constellation (GALILEO, BEIDOU...) and may be used in GPS in some specific applications replacing SBAS\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eThis research is a part of Bousnane. A.A's preparation for PHD, where Berguig. M.C is the first supervisor (for deep learning knowledge) and Dekkiche. H is the second supervisor (for GPS and geodesy knowledge)\u003c/p\u003e\n\u003cp\u003eWe, the authors of this manuscript entitled \u0026quot;[\u0026nbsp;Reducing GPS orbital and clocks errors using recurrent neural networks],\u0026quot; declare the following:\u003c/p\u003e\n\u003cp\u003eThe manuscript represents original work that has not been published previously and is not under consideration for publication elsewhere.\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no conflict of interest.\u003c/p\u003e\n\u003cp\u003eThis article does not contain any studies with human participants or animals performed by any of the authors.\u003c/p\u003e\n\u003cp\u003eThis research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.\u003c/p\u003e\n\u003cp\u003eAny data, figures, or materials taken from other sources have been properly acknowledged and cited within the manuscript.\u003c/p\u003e\n\u003cp\u003eAll GPS Data used in this work are from CDDIS archive: C. Noll, The Crustal Dynamics Data Information System: A resource to support scientific analysis using space geodesy, Advances in Space Research,Volume 45, Issue 12, 15 June 2010, Pages 1421-1440, ISSN 0273-1177, DOI: 10.1016/j.asr.2010.01.018.\u003c/p\u003e\n\u003cp\u003eAcknowledgements: We extend sincere appreciation to Dr. [Berguig. M.C] and [Dekkiche. H] for their invaluable guidance. Special thanks to [Geophysics Laboratory, University of Sciences and Technology Houari Boumediene] for their insights. Lastly, heartfelt appreciation to family and friends for unwavering encouragement. This research was made possible by their collective support and dedication.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eCho, Kyunghyun; van Merrienboer, Bart; Bahdanau, DZmitry; Bougares, Fethi; Schwenk, Holger; Bengio, Yoshua (2014). \"Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation\". arXiv:1406.1078\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eCDDIS, Crustal Dynamics Data Information System, \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://cddis.nasa.gov/index.html\u003c/span\u003e\u003cspan address=\"https://cddis.nasa.gov/index.html\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eGoogle Smartphone Decimeter Challenge, improve high precision GNSS positioning and navigation accuracy on smartphones, 2021, \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.kaggle.com/competitions/google-smartphone-decimeter-challenge\u003c/span\u003e\u003cspan address=\"https://www.kaggle.com/competitions/google-smartphone-decimeter-challenge\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eGoogle Smartphone Decimeter Challenge, improve high precision GNSS positioning and navigation accuracy on smartphones, 2022, \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.kaggle.com/competitions/smartphone-decimeter-2022\u003c/span\u003e\u003cspan address=\"https://www.kaggle.com/competitions/smartphone-decimeter-2022\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eGrossi, Enzo \u0026amp; Buscema, Massimo. 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Insights Imaging 9, 611\u0026ndash;629.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-3881623/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3881623/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eBecause satellite orbits and clocks are usually treated as known quantities in GPS positioning models, these errors are one of the most impactful in Global Navigation satellite system (GNSS) positioning, especially in single point positioning mode. In this work we tried to reduce orbit and clock errors using deep learning technique (recurrent neural network) by creating a model that predict the offsets of broadcast ephemeris and clock biases, from precise products. We tried two different methods, the first one is predicting the future offsets for the next\u0026thinsp;+\u0026thinsp;72 hours basing on both precise and broadcast ephemeris, and the second method is predicting the current offsets using only broadcast ephemeris as inputs. To create the models, we used Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) architectures. These architectures are generally used to solve sequential problems in deep learning. We got results that vary between methods in terms of satellites and time steps. For the first method we get (10-50cm) and (0.5-1.7ns) reduced from (~\u0026thinsp;130cm, ~\u0026thinsp;14ns), It increases over time steps, and we get a minimum of (44cm) for the second method.\u003c/p\u003e","manuscriptTitle":"Reducing GPS orbital and clocks errors using recurrent neural networks","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-04 11:55:53","doi":"10.21203/rs.3.rs-3881623/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"0498a3d8-db94-4429-a7da-f6047583ba33","owner":[],"postedDate":"March 4th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-03-14T17:29:31+00:00","versionOfRecord":[],"versionCreatedAt":"2024-03-04 11:55:53","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3881623","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3881623","identity":"rs-3881623","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}